The semiclassical propagator for coherent state on twisted geometry

TL;DR

This paper introduces a new set of twisted geometric variables in loop quantum gravity, enabling a semi-classical propagator formulation through a coherent state path integral that is analytically computable.

Contribution

It proposes a novel parametrization of the phase space in LQG, leading to a semi-classical propagator via a new coherent state path integral based on twisted geometry.

Findings

The new variables form a Poisson algebra similar to quantum mechanics.

The path integral can be analytically computed around complex trajectories.

The semi-classical approximation of the quantum propagator is obtained.

Abstract

A new set of twisted geometric variables is introduced to parametrize the holonomy-flux phase space in loop quantum gravity. It is verified that these new geometric variables, after symplectic reduction with respect to the Gauss constraint, form a Poisson algebra which is analogue to that in quantum mechanics. This property ensures that these new geometric variables provide a simple path measure, upon which a new formulation of coherent state path integral based on twisted geometry coherent state is established in loop quantum gravity. Especially, this path integral is analytically computable by expanding the corresponding effective action around the complex evolution trajectories at second order, and the result gives the semi-classical approximation of the quantum propagator between twisted geometry coherent state in LQG.